Mainly for home-study: Maths and empirical tools
Partial coverage for now, some revision as we apply these
Goals of this material:
Math tools you must know or learn (see handout)
(Univariate) functions, linear/nonlinear functions; the slope of a function (arc vs. point slope); concave/convex functions
Derivative of a function: a function that tells you the slope at each point
Minima, maxima
Functions of two or more variables, contour lines
(Simple) simultaneous equations
Slides, resources to help you, plus supplementary videos; www.khanacademy.org/math/
Lecture may skip to Mini-lecture: Empirical microeconomics/econometrics here
Slope of \(y = f(x)\)
The change in y for a given change in x. ‘Rise over run’ \((\Delta y / \Delta x)\).
Nonlinear (univariate) function : A function \(f(x)\) of a form other than \(f(x) = y=a+bX\);
For linear functions the slope is the same at any point. For nonlinear functions it may differ at each point.
Convex function: Slope everywhere increasing, unique minimum where slope \(=0\) Concave function: Slope everywhere decreasing, unique maximum where slope \(=0\)
\(f'(x)\) tells us the (point) slope of the function \(f(x)\) at any point \(x\).
Derivative of \(f(x) = x^2 -4x + 1\) is \(f'(x) = 2x - 4\)
E.g., slope at \(x=1\) is \(f'(x;x=1) = 2\times1 - 4 = -2\)
The slope is zero where \(f'(x)=2x-4=0\), or where \(x=2\)
Is \(x=2\) at a min, a max, or neither? How do we know?
\(f'(x)\) is a function that tells us the slope of \(f(x)\), or how \(f(x)\) changes in \(x\) at any point \(x\)
Oversimplifying:
slope always increasing \(\rightarrow\) \(f''(x)>0\) everywhere \(\rightarrow\) convex (u-shaped) function \(\rightarrow\) single where \(f'(x)=0\)
slope always decreasing \(\rightarrow\) \(f''(x)<0\) everywhere \(\rightarrow\) concave (inverse-u) fncn \(\rightarrow\) single where \(f'(x)=0\)
Utility, profit, cost, production, returns, etc.
\[y=f(x,z)\]
\[y=f(x,z)\]
\(y\) may increase and/or decrease in \(x\) and in \(z\),
The rate of increase of y in \(x\) may depend on the values of \(x\) and \(z\)
E.g., \[y=\sqrt(xz) = x^{1/2}z^{1/2}, x \geq 0, z \geq 0\]
Projecting a function up from X,Y space into the Z axis:
We will come back to this later!
Level sets: E.g., indifference curves, isoquants and isocost curves.
Contour lines on a map
Consider a production function:
\[Y = f(K,L) = \sqrt(KL)\]
Setting this equal to 1 we can map out ‘all combinations of K and L that produce output \(Y=1\)’:
\[ Y = \sqrt(KL) = 1 \rightarrow KL = 1 \]
\[ \rightarrow K = 1/L \]
Setting this at Y = 2
\[ Y = \sqrt(KL) = 2 \rightarrow KL = 4 \] \[ \rightarrow K = 4/L \]
E.g.,
\[ X + Y = 3 \] \[ X - Y = 1 \]
Holds only where \(X=2, Y=1\), the ‘unique solution’
Trying to estimate demand curve, hypothesize linear function \[Q_d = a-bp\]
Suppose we know price is shifting because of costs, shifts in supply curve, or the firm is experimenting. Observe price & quantity data for a period where ceteris paribus is reasonable.
Fit ‘best’ line (minimise error) through these points
Estimate demand slope & intercept, use to make inferences
Never fits exactly. why not?
All [most] economic theories employ the assumption that ‘other things are held constant.’
the points may lie on several different demand curves, and attempting to force them into a single curve would be a mistake.
\(\rightarrow\) Carefully ‘control’ for other observable factors (a partial solution)
Empirical work has estimated:
\[ Q = 85 - 0.4P \: (D) \] \[ Q = 55 + 0.6P \: (S) \]
Solving: \(85 - 0.4P=55+0.6P \rightarrow P = 30, Q = 73\)
Approximates 2000-2002 price
Why did price rise to US$130 in 2008 and fall to $50 by March ’09?
\[Q_D = 112 - 0.4P\] \[Q_S = 55 + 0.6P\]
See Handout/web-book
Representative answers for each problem set given about 1-week after posting
Five support classes (tutorials), cover parts of these
E.g., …
E.g., “True or false: It is valid to plot observed prices and quantities traded in a market and fit a line through them to estimate a market demand curve.”
(NS - Chapter 2)
Consider a decision you recently made?
Define this decision clearly.
How do you think you decided among these options?
2 minutes: discuss with your neighbour
Suppose I asked you
‘State a rule that governs (determines/characterizes) how people do make decisions’…
I want this rule to be…
Informative (it rules out at least some sets of choices)
Predictive (people rarely if ever violate this rule)
Similar question:
‘State a rule that governs how people should make decisions’..
By ‘should’ I mean that they will not regret having made decisions in this way.
2 minutes: discuss with your neighbour
If people did follow these rules, what would this imply and predict?
Rules defined as ‘axioms about preferences’